Some Analytical Aspects of the Theory of Integrable Systems

Alexander R. Its的研究项目集中在与黎曼-希尔伯特和同构方法相关的可积系统理论的分析方面。研究的问题包括精确可解量子场和统计力学模型的相关函数的渐近分析，随机矩阵和正交多项式理论中的双标度极限的研究，以及Painleve和KdV型可积微分方程解的全局渐近分析。可积系统理论是一个不断扩大的领域，它作为现代数学和理论物理的许多分支的新的分析和代数思想的主要来源之一，发挥着越来越重要的作用。同时，它为研究在现代非线性科学和技术的许多应用中起核心作用的一些基本数学模型提供了一种有效的分析工具。具体而言，上述研究方向涉及的数学模型构成了以下领域的理论基础：凝聚态物质、高能和等离子体物理、非线性流体力学和高比特率电信系统。

DMS-9801608 Alexander R. Its The research project is focused on the analytical aspects of the theory of integrable systems related to the Riemann-Hilbert and isomonodromy methods. The problems under consideration include the asymptotic analysis of the correlation functions of exactly solvable quantum field and statistical mechanics models, the investigation of the double scaling limits in the theory of random matrices and orthogonal polynomials, and the global asymptotic analysis of the solutions of integrable differential equations of the Painleve' and KdV types. The theory of integrable systems is an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. Simultaneously, it provides an efficient analytic tool for the study of some of the fundamental mathematical models which play a central role in numerous applications arising in modern nonlinear science and technology. Specifically, the research directions indicated above deal with the mathematical models which form the theoretical basis for the following fields: condensed matter, high energy and plasma physics, nonlinear hydrodynamics, and high bit-rate telecommunication systems.